gausslemma2d - Metamath Proof Explorer

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Theorem gausslemma2d 25958

Description: Gauss' Lemma (see also theorem 9.6 in [ApostolNT] p. 182) for integer 2: Let p be an odd prime. Let S={2,4,6,...,(p-1)}. Let n denote the number of elements of S whose least positive residue modulo p is greater than p/2. Then ( 2 | p ) = (-1)^n. (Contributed by AV, 14-Jul-2021.)

**Hypotheses**
Ref	Expression
gausslemma2d.p	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
gausslemma2d.h	⊢ 𝐻 = ((𝑃 − 1) / 2)
gausslemma2d.r	⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
gausslemma2d.m	⊢ 𝑀 = (⌊‘(𝑃 / 4))
gausslemma2d.n	⊢ 𝑁 = (𝐻 − 𝑀)

**Assertion**
Ref	Expression
gausslemma2d	⊢ (𝜑 → (2 /_L 𝑃) = (-1↑𝑁))

Distinct variable groups: 𝑥,𝐻 𝑥,𝑃 𝜑,𝑥 𝑥,𝑀 Allowed substitution hints: 𝑅(𝑥) 𝑁(𝑥)

**Proof of Theorem gausslemma2d**
Step	Hyp	Ref	Expression
1		gausslemma2d.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
2		gausslemma2d.h	. . 3 ⊢ 𝐻 = ((𝑃 − 1) / 2)
3		gausslemma2d.r	. . 3 ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
4		gausslemma2d.m	. . 3 ⊢ 𝑀 = (⌊‘(𝑃 / 4))
5		gausslemma2d.n	. . 3 ⊢ 𝑁 = (𝐻 − 𝑀)
6	1, 2, 3, 4, 5	gausslemma2dlem7 25957	. 2 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)
7		eldifi 4054	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
8		prmnn 16008	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	8	nnred 11640	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
10		prmgt1 16031	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
11	9, 10	jca 515	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
12	1, 7, 11	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
13		1mod 13266	. . . . . 6 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → (1 mod 𝑃) = 1)
15	14	eqcomd 2804	. . . 4 ⊢ (𝜑 → 1 = (1 mod 𝑃))
16	15	eqeq2d 2809	. . 3 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1 ↔ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)))
17		neg1z 12006	. . . . . . . . . . 11 ⊢ -1 ∈ ℤ
18	1, 4, 2, 5	gausslemma2dlem0h 25947	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
19		zexpcl 13440	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℤ)
20	17, 18, 19	sylancr 590	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ)
21		2nn 11698	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℕ)
23	1, 2	gausslemma2dlem0b 25941	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ ℕ)
24	23	nnnn0d 11943	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
25	22, 24	nnexpcld 13602	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑𝐻) ∈ ℕ)
26	25	nnzd 12074	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ)
27	20, 26	zmulcld 12081	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ)
28	27	zred 12075	. . . . . . . 8 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℝ)
29		1red 10631	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
30	28, 29	jca 515	. . . . . . 7 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ))
31	30	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ))
32	1	gausslemma2dlem0a 25940	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
33	32	nnrpd 12417	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
34	20, 33	jca 515	. . . . . . 7 ⊢ (𝜑 → ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺))
35	34	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺))
36		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃))
37		modmul1 13287	. . . . . 6 ⊢ (((((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ) ∧ ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺) ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃))
38	31, 35, 36, 37	syl3anc 1368	. . . . 5 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃))
39	38	ex 416	. . . 4 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃)))
40	20	zcnd 12076	. . . . . . . . 9 ⊢ (𝜑 → (-1↑𝑁) ∈ ℂ)
41	25	nncnd 11641	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐻) ∈ ℂ)
42	40, 41, 40	mul32d 10839	. . . . . . . 8 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) = (((-1↑𝑁) · (-1↑𝑁)) · (2↑𝐻)))
43	18	nn0cnd 11945	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
44	43	2timesd 11868	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
45	44	eqcomd 2804	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 𝑁) = (2 · 𝑁))
46	45	oveq2d 7151	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝑁 + 𝑁)) = (-1↑(2 · 𝑁)))
47		neg1cn 11739	. . . . . . . . . . . 12 ⊢ -1 ∈ ℂ
48	47	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → -1 ∈ ℂ)
49	48, 18, 18	expaddd 13508	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁)))
50	18	nn0zd 12073	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
51		m1expeven 13472	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(2 · 𝑁)) = 1)
53	46, 49, 52	3eqtr3d 2841	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑𝑁) · (-1↑𝑁)) = 1)
54	53	oveq1d 7150	. . . . . . . 8 ⊢ (𝜑 → (((-1↑𝑁) · (-1↑𝑁)) · (2↑𝐻)) = (1 · (2↑𝐻)))
55	41	mulid2d 10648	. . . . . . . 8 ⊢ (𝜑 → (1 · (2↑𝐻)) = (2↑𝐻))
56	42, 54, 55	3eqtrd 2837	. . . . . . 7 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) = (2↑𝐻))
57	56	oveq1d 7150	. . . . . 6 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((2↑𝐻) mod 𝑃))
58	40	mulid2d 10648	. . . . . . 7 ⊢ (𝜑 → (1 · (-1↑𝑁)) = (-1↑𝑁))
59	58	oveq1d 7150	. . . . . 6 ⊢ (𝜑 → ((1 · (-1↑𝑁)) mod 𝑃) = ((-1↑𝑁) mod 𝑃))
60	57, 59	eqeq12d 2814	. . . . 5 ⊢ (𝜑 → (((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃) ↔ ((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃)))
61	2	oveq2i 7146	. . . . . . . 8 ⊢ (2↑𝐻) = (2↑((𝑃 − 1) / 2))
62	61	oveq1i 7145	. . . . . . 7 ⊢ ((2↑𝐻) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃)
63	62	eqeq1i 2803	. . . . . 6 ⊢ (((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃) ↔ ((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃))
64		2z 12002	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
65		lgsvalmod 25900	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((2 /_L 𝑃) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃))
66	64, 1, 65	sylancr 590	. . . . . . . . 9 ⊢ (𝜑 → ((2 /_L 𝑃) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃))
67	66	eqcomd 2804	. . . . . . . 8 ⊢ (𝜑 → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((2 /_L 𝑃) mod 𝑃))
68	67	eqeq1d 2800	. . . . . . 7 ⊢ (𝜑 → (((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃) ↔ ((2 /_L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃)))
69	1, 4, 2, 5	gausslemma2dlem0i 25948	. . . . . . 7 ⊢ (𝜑 → (((2 /_L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
70	68, 69	sylbid 243	. . . . . 6 ⊢ (𝜑 → (((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
71	63, 70	syl5bi 245	. . . . 5 ⊢ (𝜑 → (((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
72	60, 71	sylbid 243	. . . 4 ⊢ (𝜑 → (((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
73	39, 72	syld 47	. . 3 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
74	16, 73	sylbid 243	. 2 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1 → (2 /_L 𝑃) = (-1↑𝑁)))
75	6, 74	mpd 15	1 ⊢ (𝜑 → (2 /_L 𝑃) = (-1↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ifcif 4425 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 4c4 11682 ℕ₀cn0 11885 ℤcz 11969 ℝ⁺crp 12377 ...cfz 12885 ⌊cfl 13155 mod cmo 13232 ↑cexp 13425 ℙcprime 16005 /_L clgs 25878

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-ioo 12730 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-prod 15252 df-dvds 15600 df-gcd 15834 df-prm 16006 df-phi 16093 df-pc 16164 df-lgs 25879

This theorem is referenced by: 2lgs 25991

W3C validator